1/2x+1/3x-2/3x=7

Solution for 1/2x+1/3x-2/3x=7 equation:

1/2x+1/3x-2/3x=7

We move all terms to the left:

1/2x+1/3x-2/3x-(7)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R



Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We calculate fractions


3x/6x^2+(-4x+1)/6x^2-7=0

We multiply all the terms by the denominator


3x+(-4x+1)-7*6x^2=0

Wy multiply elements

-42x^2+3x+(-4x+1)=0

We get rid of parentheses

-42x^2+3x-4x+1=0

We add all the numbers together, and all the variables

-42x^2-1x+1=0

a = -42; b = -1; c = +1;
Δ = b2-4ac
Δ = -12-4·(-42)·1
Δ = 169
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{169}=13$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1)-13}{2*-42}=\frac{-12}{-84}
=1/7
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1)+13}{2*-42}=\frac{14}{-84}
=-1/6
$